Construction of PDF-based imaging phenotypes. Consider MRI scans for \(n\)  LGG subjects across all four imaging sequences and the corresponding tumor segmentation masks. Let \(N_i\)  denote the number of voxels in the tumor region for each subject \(i\) . Let \(\mbox{{$\mathbf {c}$}}_i\)  denote array indices of the mid-point for the minimal bounding box of the tumor region, and \(\tau \in \mathbb {N}\)  denote the number of spherical shells we divide the tumor region into. In Algorithm S1 of Section S1, we describe construction of kernel density estimates \(f^{(t,m)}_i\)  corresponding to spherical shell \(t \in \lbrace 1,\ldots ,\tau \rbrace \)  from MRI sequence \(m \in M = \lbrace T1,T1Gd,T2,FLAIR\rbrace \)  for subject \(i\) . We divide the tumor region into \(\tau \)  spherical shells based on the total number of voxels in the tumor region, so that we have the same number of spherical shells for all subjects. While the innermost layer is a sphere, we refer to all of the \(\tau \)  layers as spherical shells or layers hereafter. Hence, for each subject, we have the tumor region divided into \(\tau \)  spherical shells across all four imaging sequences. This is represented in panel A of Figure REF , with spherical layers from four MRI sequences and their corresponding PDFs. Via Algorithm S1, we obtain kernel density estimates \(f^{(t,m)}_i\)  based on the voxels from MRI sequence \(m\)  in spherical shell \(D^{(t)}_i\)  for subject \(i\) . Using a Riemannian-geometric framework for analyzing PDFs [1]}, we implement a principal component analysis (PCA) at each layer \(t=1,\ldots ,\tau \) , which generates layer-specific PC scores (as shown in Figure REF ). The geometric framework for PDFs is necessary in order to ensure that the PCA is compatible with the nonlinear structure of the space of the voxel-intensity PDFs, and has been successfully used in several works [2]}, [3]}. For ease of exposition, we omit the mathematical details here and describe them in Section S1.2.
